![\"\"](\"/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=34be7fb067b94bd17e8baea5428d5408.png\")
having vertical and horizontal asymptotes.(a)
\Consider a graph of the function having vertical and horizontal asymptotes.
Graph :
\Graph the function , it has a vertical aymptote as well as horizontal asymptote.
![\"\"](\"/userfiles/234-2a.gif\")
Observe the graph.
\The graph of the function ![\"\"](\"http://www.mathskey.com/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=34be7fb067b94bd17e8baea5428d5408.png\")
intersects the horizontal asymptote but it does not intersects the vertical asymptote.
Therefore the graph of the function may intersects the horizontal asymptotes but it does not intersects the vertical asymptotes.
(b)
\The function can have no, one or two horizontal asymptotes but it does not have more than two horizontal asymptotes.
The function crosses ![\"\"](\"http://www.mathskey.com/test/fckeditor/editor/plugins/fckeditor_wiris/integration/showimage.php?formula=8da2268e35b6b6daab9c47f71c4c8bf4.png\")
-axis only one ones, if it touches more than ones then it is not a function.
Graph :
\Graph the function :
The graph of the function having no horizontal asymptote :
\![\"\"](\"/userfiles/234-2b1.gif\")
The graph of the function having a only one horizontal asymptote :
\![\"\"](\"/userfiles/234-2b2.gif\")
The graph of the function having two horizontal asymptote :
\![\"\"](\"/userfiles/234-2b3.gif\")
(a)
\Graph of the function may intersects the horizontal asymptotes but it cannot intersects the vertical asymptotes.
(b) Graph of the function .
The graph of the function having no horizontal asymptote is
\
The graph of the function having only one horizontal asymptote is
\
The graph of the function having two horizontal asymptotes is
\.